
In this section, we aim to provide the computational complexity of the polyblock algorithm,
which significantly depends on the structure of the objective function and the constraints that
form the normal set. We note that the proposed algorithm includes the following steps. In the first step, we obtain the best vertex with its projection belonging to the normal set. Next, we
obtain the projection of the selected vertex. Finally, having the improper vertices removed, the
new vertex set is found.
We assume that the dimension of the optimization problem is \(L_{1}\) , the
number of iterations in the overall polyblock algorithm for convergence is \(L_{2}\)  while the number
of iterations for the projection of each vertex is \(L_{3}\) . Then, the complexity order can be expressed as \(\mathcal {O}(L_{2}(L_{2}\times L_{1}+L_{3}))\)  [1]}, [2]}.
[t]
Outer Poly-block Approximation Algorithm
[1]


An function \(f(\cdot )\) , a compact normal set \(\mathcal {G}\)  and a closed conormal set \(\mathcal {H}\) , such that \(\mathcal {G}\bigcap \mathcal {H}\ne {Ø}\) 
Output: an \(\epsilon \) -optimal solution \(\mathbf {x}^*\) 
Initialize iteration index \(l=0\) , poly-block \(\mathbf {\Omega }^{(1)}\)  be box \([0,\mathbf {b}]\)  that encloses \(\mathcal {G}\bigcap \mathcal {H}\)  with vertex set \(\omega ^{(1)}=\lbrace \mathbf {b}\rbrace \) . \(\epsilon \)  is a small positive number. \(V^{(l)}=-\infty \) 
Repeat
